Radiative capture of low-energy neutrons in the shell-model approach to nuclear reactions

1

~

NuclearPhysics A215 (1973) 605--616; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher

RADIATIVE CAPTURE OF LOW-ENERGY NEUTRONS IN T H E S H E L L - M O D E L A P P R O A C H T O N U C L E A R R E A C T I O N S E. BORIDY t Theoretical Physics, University of LiJge, 4000-LiJge I, Belgium and C. MAHAUX *t Physics Department, University of Rochester, Rochester, New York Received 2 July 1973 Abstract: The shell-model approach to photonuclear reactions is applied to the radiative capture of low-energy neutrons. The direct, compound and channel resonant contributions to the collision matrix appear in a natural way. We show that the experimental data in the vicinity of the 3s giant resonance can be quantitatively interpreted in terms of the channel resonant capture. The relationship with the R-matrix approach of Lane and Lynn is discussed. 1. Introduction A theory of radiative capture of low-energy neutrons has been developed by Lane and Lynn 1-3) in the framework of R-matrix theory. In that approach, a formal summation over distant R-matrix levels is performed in order to introduce the average shell-model potential. The latter plays a fundamental role in the qualitative and quantitative interpretation of some phenomena which have been recently discovered in radiative capture reactions 4-6). It appears, therefore, simpler to use the shellmodel approach to nuclear reactions 7), which includes from the start the average nuclear field. The extension of this approach to photon channels was performed in ref. 8). In the present paper, we apply the results obtained in ref. 8) to the radiative capture of low-energy neutrons. In sect. 2 we briefly recall and give more explicit expressions for some results obtained in ref. 8). We compare in sect. 3 these formulae with those derived from R-matrix theory. In sect. 4, we construct a channel capture model and we demonstrate that it is in quantitative agreement with experiment, for neutron capture in the mass region A ~ 60. 2. Basic equations The nuclear Hamiltonian is divided into a shell-model Hamiltonian H0 and a residual interaction V. Let Z~ (c = 1. . . . , A) be the eigenstates (channel states) belonging to the continuous spectrum of Ho and ~bi ( j = 0 , . . . , M ) the bound eigent Post-doctoral fellow, National Research ~Council of Canada, ,t On leave from the University of Liege, Belgium. 6O5

606

E. BORIDY AND C. MAHAUX

states of Ho. The element of the collision matrix between a particle channel c and a photon channel p is given by 8) !

M

S~p = S~,m'+ Z YacYap , ~=o E - E a +½iF ~

(2.1)

where the smoothly energy-varying term S~pm" reads (we neglect the direct channelchannel coupling): $.m, = Scp 2np~:(°)~plH~.tlz~). (2.2) The energy density of photon states is denoted by p and H~.t represents the interaction between the nucleons and the electromagnetic field. The state o)~pp is a product of a bound nuclear state by a photon state vector IP). The quantity y~ in eq. (2.1) is the partial width amplitude in particle channel c, while Yap is the corresponding quantity in the photon channel p. We have s) 7a~ -- (2n)~(z~lVlaa), ]lAp ~

, ( 1 ) ..t-

gap t

(2.3)

,(2)

(2.4)

l,a.p ~'

where (2.5)

y~) -- (2ztp)~(°)kU, IHi,t[Qa) , =

de'

~'= 1.,~o,

1

E + - E ' <(1)~plH~"tlz[">Ya~'(E')"

(2.6)

In eqs. (2.3) and (2.5), the compound nuclear state f2A is a linear combination of the bound eigenstates ~bj, while ec, denotes the threshold energy in channel c'. In the j-j coupling scheme, the S-matrix element for the radiative capture reaction N + A ~ B + 7 reads (2.7) S c p ~ ',.~s) 3 ~ x I J ",.,,~,I l a ze,, where j = l+½,

(2.8)

j+IA = J = IB+~.

Here, ] is the total angular momentum obtained by coupling, in the entrance channel, the total angular momentumj of the incident nucleon to the spin IA of the target or, in the exit channel, the spin IB of the residual nucleus to the total angular momentum of the photon. The labels ~^ and ~B in eq. (2.7) represent additional quantum numbers necessary to specify completely the initial and final nuclear states, and n denotes the parity of the emitted photon. The integrated cross section corresponding to the the radiative capture process is given by 7z

2J+ 1

-~s)

2

(2.9)

RADIATIVE CAPTURE

607

where k~ is the wave number of the incident nucleon. In the following, we use for the shell-model potential a Woods-Saxon form with spin-orbit coupling:

( h 2 12 d f(r)l , " l, Vo = - - V o f ( r ) - V s . o . x ~ / [~ drr

(2.10)

where m~ is the pion mass, and

f(r) = [1 +exp (r-R)/a] -1.

(2.11)

Expression (2.2) for S~vm" shows that this part of the collision matrix corresponds to a "direct" capture, while the second term on the right-hand side of eq. (2.1) represents the "resonance capture". Using eqs. (2.4)-(2.6), one can write M

Scp :

cv -I- X-" L=o S .....

,

,(1)

ra~rap E-Ea+½iF a

M

,

..}_A=EO

,(2)

~acrap

= E-Ex+½iF a

(2.12)

The second term on the right-hand side of this equation can be called the "compound '(~) involves the emission from the compound nuclear resonance capture" since rap nuclear state t2a. In 7~2), the photon is emitted by the channel configuration, and the last term of eq. (2.12) can therefore be referred to as the "channel resonance capture". 3. Comparison with the R-matrix approach it is useful to discuss the relationship between the R-matrix 1-3, 5, 6) and the shell-model approaches to photonuclear reactions, since the experimental data have, up to now, been analyzed in the framework of R,matrix theory. In the R-matrix theory, a channel radius a c is introduced, which separates the internal from the external region. In the internal region, one defines a discrete set of eigenstates {Xa} of the nuclear Hamiltonian. The zero photon part of the total wave function is written, in the internal region, in the form

( o ) ~ = ~ Q](E)Xa,

(3.1)

A

where the coefficients Qa( CE ) are determined by the condition that ( o ) ~ and its derivative are continuous at the surface of the internal region. One then obtains an expression of the collision matrix element Sop similar to that given by eq. (2,12) with the difference that S~j,m" is replaced by Sc~. The latter quantity corresponds to the "hard sphere" capture; it is given by eq. (2.2) with X~ replaced by the hard sphere wave function o~(HS) ~o~(HS) oc .~y ~ - e - 2i,c~ c~c = 0

(re > = ac) (re < ac),

(3.2)

where ~¢¢ and d~ are incoming and outgoing waves, and tr¢ is the hard sphere scattering phase shift. A term analogous to the second term on the right-hand side of eq. (2.12)

608

E. BORIDY AND C. MAHAUX

also appears in R-matrix theory and is called the "internal resonance capture", since the states Xx of eq. (3.1) (which are the analogues of our f2x) are defined only in the internal region. The equivalent of the last term of eq. (2.12) (corresponding to the channel resonance capture) arises from the contribution of the external region, other than the hard sphere term. While these three terms are formally similar to those appearing in eq. (2.12), they are not identical to them. For instance, the hard sphere component does not represent the direct process since co~(HS) ignores the average nuclear interaction between the projectile and the target. Lane and Lynn 1- 3) have shown how one can extract from the resonance terms the contribution of the "faraway" resonances. If this contribution is added to S~s, one obtains the "direct capture" component of Sop. The remaining resonant part of Sep contains only the contribution of "local" resonances. The extraction of the contribution of the distant resonances, as shown by Lane and Lynn, leads to an expression similar to that of s.m. our term S~p with, however, a scattering neutron wave function obtained from a complex potential. We note that the introduction of a complex potential does not appear to be straightforward since no loss of flux takes place for low-energy neutrons and since the formula refers to data without any averaging over energy. In a forthcoming paper, we discuss that, for the calculation of most quantities of interest, a summation over the distant resonances is usually not required in the shell-model approach since one introduces, from the beginning, an average nuclear potential. We note, however, that eqs. (2.5) and (2.6) imply that the partial width amplitudes ?~p and ?~¢ are correlated s). Thus, the sum over the distant resonant terms in eq. (2.1) does not vanish, and contributes to the "background" cross section 5). We have shown in ref. 9) that this contribution is usually small compared with the sum of direct capture contribution.

4. Applications to low-energy neutron capture We first construct a simple model for the channel capture. We then show that this simple model gives results which are in fair agreement with experiment, for several targets with A ~ 60. 4.1. A SIMPLE MODEL FOR THE CHANNEL CAPTURE We consider the model based on the following three assumptions. (i) The electromagnetic decay of the compound nuclear states to channel p is weak, i.e.

(4.1) ~2p" The justification of this assumption will be given a posteriori, by comparing its predictions with experiment (subsect. 4.3). (ii) There exists one isolated resonance 2. (iii) There exists only one particle (entrance) channel c. The collision matrix element then reads ./(~, << / ',(2) .., gp ~

,

,(2)

,.., s.m. ~xcr~.p Scp ~ Sop + E - E x + ½ i F x

.

(4.2)

R A D I A T I V E CAPTURE

609

We separate ¥2p ,(2) into its real and imaginary parts: yap'(2)= p,rp

•,~o

dE'

1 E-E'

(O)~PplHintlZ[,)y,~c(E')- ircp½((t)Y"plHintlX~E)~',~c(E).

(4.3)

The main contribution to the principal value integral in eq. (4.3) requires the knowledge of the particle partial width amplitude in the domain E' ~ E. Let us assume

/

\

o

(c)

/ ~

"'

.......

!"!

", . . . . . . . . . . . . .

0

..*--°'""""

10

15

20

r (fm)

Fig. 1. The radial continuum s-wave function at 98.1 keV (curve a), the radial 2p~ bound wave function (curve b) and the overlap integral f(r) of eq. (4.20) (curve c), for 6°Ni.

30

20

O .13 L

o

~o (.O

60N/

,Fe

10/ 0

• 10-6

I

i

10-~'

i

i

10-2

\

i

i

10o

I

10 2

£ (MeV)

Fig. 2. The dependence off(e', eo ) versus e" (eq. (4.4)).

610

E. B O R I D Y A N D C. M A H A U X

that we can use the following approximation, whenever Z~ appears in a matrix element 7, lo, 11):

Z~ ~ f(e, eo)Z~o,

(4.4)

where e = E - e c , and Z~o is the channel wave function at an energy E o = ec+e o, lying in the energy domain of interest. We have then

),;,c(E')

=

f(e', eo)7~¢(Eo).

Eq. (4.3) can now be written

_i~p½((1)~,p[Hi,tlZ~>),x~(E )

(2)= ~P

(4.5)

El+iReF(e,

(4.6a)

izcf2(e, So).

(4.6b)

Im

F(e, eo)1 eo)J '

where t'o

f2(e',

eo) e--e'

F(e, eo) = P / Jo

de'-

s S

. 2.0

.

. ~._

."

. ,

.

1"

.!"

2 ~.

.-"

E -4

LL

10

-11

v °~ -6

0 10~I

t 100

I 101

10 2

E/E O

Fig. 3. Variation o f the ratio of the real and imaginary parts o f

F(e, eo)

(eq. (4.6b)), with energy e.

The function f(e', eo) can be determined empirically by computing the radial part E, (see eq. (4.11) below) of the channel wave function g~,. In fig. 1, we show the wave Uc E' function uc, at 98.1 keV for the target nucleus 6°Ni. The parameters of the WoodsSaxon potential have been taken from ref. 12), namely tlo = 42.8 MeV, Vs.o. = 9.31 MeV, R = 1.3 A ~ fro, a = 0.69 fm. We checked that approximation (4.4) is indeed justified for 0 < e' < 5 MeV. The dependence o f f ( e ' , eo) on e' is shown in fig. 2. The decrease o f f ( E , e0) when e' ~ 0 is due to penetration effects, while that above ,~ 30 keV is related to the fact that the potential is almost critical, i.e. is close to the one which has a bound state with vanishing binding energy. For 56Fe and 6°Ni

RADIATIVE CAPTURE

611

target nuclei, the functionf(s', So) can be parametrized in the form 3.25

f(s', So) ~ (log s ' - l o g So)2+3.25 '

(4.7)

with s o equal to 31 and 25 keV, respectively. With the energy dependence (4.7), the real and imaginary parts of F(s, So) can be easily calculated. The results shown in fig. 3 indicate that, for incident energies s below %, the real part of F(8, %) is larger in absolute magnitude than the imaginary part. In particular, the latter approaches zero when s ~ 0, while the real part tends towards a finite value. For s > 4.5 eo, however, the ratio Im FIRe Fis larger than two. 4.2. THE PARTIAL WIDTH IN THE PHOTON CHANNEL From eqs. (4.1) and (4.6) we find the following expression for the photon width:

Irzpl

= I~zpl2 =

~Zpl<(1)~,plHintlZ~>12rac(1 + ~z),

(4.8)

where Fac is the partial width in the entrance channel, and

0~ -- Re F(s~, So) Im t(sx, s0)"

(4.9)

For an electric dipole radiation, we have, for unpolarized projectile and target a), /,p

a 2 3 = v~r k p r a c ( l + ~

z I(~'p(Ia)llQdlz~(J))l )

2

2J+ 1

,

(4.10)

where kp is the wave number of the emitted photon (kp = Ep/hc), and Q1 the electric dipole operator. For a given state (lj) of the projectile, the wave function Z~(J) of the system in its initial state can be written, ignoring antisymmetrization,

X~ - u~(r) l(/½)J, IA; J ) , /.

(4.11)

where we have isolated the radial wave function of the incident neutron. For the final state wave function, we assume, following Lane and Lynn ~-3), that the incident neutron is captured into a low-lying single-particle final state (nTj'), leaving the core of the other nucleons undisturbed. We can then write ~Vp(s.p.) = Z a('~) wn'rJ'(r)

t]n'l'j

n'l'j"

[(/'½)J', IA; IB),

(4.12)

r

where "atzB) n ' l ' j ' is the amplitude of the corresponding configuration (n'l~j'). Its square is the familiar spectroscopic factor. It should be noted that the radial wave function utjr (r) is normalized to the following asymptotic behaviour:

~, , ~[ 2# ~½ . u'jtr),2o~ ! ~ - k ) sm (kr-½1rc +tS,j),

(4.13)

612

E. BORIDY AND C. MAHAUX

where 6tj is the nuclear phase shift due to the shell-model potential Vo, # is the reduced mass of the projectile-target system and k is the wave number of the incident neutron with energy ~ = E - e ~ . From eqs. (4.10)-(4.12), we obtain the following expression for the radiative partial width: flap = 98-/zZkae2(1 + ~2)-/'ac Z LUn,l,j,..I rat1.)]2/-atn,l,j,"

nlAZaJ,

lj~tljl'j"

(4.14)

n'|'j"

with

i,,vj,.tj = l ffdrwn,vy(r)ru~(r ) 2,

(4.15)

AtA,Bs _ 1<(l'½)j', I^; IallYall(lk)j, IA; d>l 2

(4.16)

ljl'j"

--

2J+l

= 3 (2IB + 1)(2j+ 1)(2j'+ 1)(2/+ 1) t+/g

x W2(jJjTa; IAl)WZ(ljlT'; ½1)(llOOll'O)2,

(4.17)

where the W are Racah coefficients. Below, the effective charge ~ of the neutron is taken equal to - eZ[A. 4.3. RESULTS The simple model of subsect. 4.1 leads to eq. (4.14) which states that the photon and neutron partial widths are fully correlated. It is apparent from eqs. (4.1) and (4.8) that this model may be valid at best whenever S~j,m" is large. This occurs (i) when the spectroscopic factor tv,,vj, r#~B) j-12 is large, and (ii) when the incident neutron energy E is close to a single-particle resonance in channel c, since then the dipole overlap integral I,,r~, ' zj is large. In practice, correlations between partial widths in neutron and photon channels have been found 5) in the mass regions 32 < A < 60, 90 A < 110, 160 < A < 175. It is shown in ref. a) that in these mass regions, the matrix elements I show very strong peaks, for the 3s -~ 2p, 3p ~ 3s and 4s ~ 3p transitions, respectively. We have performed numerical calculations for the target nuclei 6°Ni and 56Fe, which fall in the mass region A ~ 60, where correlations are observed. Jackson and Strait 13) have measured the photon partial widths of several resonances appearing in the 61Ni(~, n)6°Ni and S7Fe(]), n)S6Fe reactions. We have used these data to test our simple model for s-wave capture near the 3s giant resonance. The neutron partial widths Fac are given by Garg et al. x4), and the spectroscopic factors 02 have been taken from ref. as). Photon partial widths have been computed using eq. (4.14), with the potential well parameters of ref. ~2). Results for seven resonances in 6°Ni (n, 7)61Ni are shown in table 1, where we give the quantity Irapl. In all these cases, (which is a function of Ea) is about unity. In view of the large uncertainties in the

RADIATIVE C A P T U R E

613

experimental neutron partial widths [compare, for instance, the results of Garg et aL 14) with those of Bilpuch et aL 16)], in the spectroscopic factors, and in the experimental photon partial widths (+20 %), the agreement between our model and experiment may be considered rather satisfactory. TABLE 1 Comparison of predicted and measured photon partial widths for some resonances in ~°Ni(n, 9')5 9Ni E~ (keV)

I/~l (eV)

r ~
12.47 43.08 98.10 107.80 162.10 186.50 198.00

1.914-0.06 0.144-0.03 1.074-0.16 1.75 5.30 4- 2.00 5.70 4-2.30 3.504-2.30

theory

F~., (eV) exp. ref. la)

I'a, (eV) ref. XT)

0.556 0.021 0.103 0.172 0.422 0.422 0.252

0.367 0.018 0.102 0.209 0.166 0.062 0.557

0.390 0.006 0.045 0.030 0.051 0.227 0.114

TABLE 2 Comparison of predicted and measured photon partial widths for some resonances in 5~Fe(n, 7)57Fe E~ (keV) 27.9 74.0 123.5 130.0 141.0 169.0 188.0

rac (keV)

IFa,l (eV) theory

F2, (eV) exp ref. xa) ref. z2)

1.52 4-0.04 0.54 4-0.07 0.0144-0.005 0.66 ±01.10 2.27 4-0.20 0.76 4-0.11 3.20 4-0.23

0.565 0.129 0.003 0.117 0.387 0.117 0.465

0.112 0.082 0.119 0.105 0.068 0.066 0.423

0.17 0.24 0.056 0.072 0.12 0.42

-P;.n (eV) ref. XT) 0.127 0.021 0.006 0.018 0.081 0.022 0.096

The same conclusions apply to the results of 56Fe(n, y)STFe presented in table 2. Bhat et aL 17) used the R-matrix valency nucleon model of Lynn 3) to calculate the resonance photon partial widths for 6°Ni and 56Fe. Their results are also shown in tables 1 and 2. We see that both shell-model and R-matrix approaches give an orderof-magnitude agreement with the experimental data. This is remarkable, in view of the simplicity of these models. Since the potential is close to a critical one, the wave function u~ is almost independent of r for R < r < 2R, where R is the nuclear radius (see fig. 1). This may explain why the R-matrix calculations are successful, when the single-particle estimate (½h2/Ma2) is used for the reduced width, since this value is based upon the assumption that u is independent of r. We have also plotted in fig. 1 the value of the 2p~ bound state wave function (curve b) and of the overlap integral

(curve c),

/(0 ;

I:

(4.18)

614

E. BORIDY A N D C. M A H A U X

for 6°Ni, at ~ = 98.1 keV. For r ~ 1.3A ~ ~ 5 fln, the integral f(r) is negligibly small. At r ~-, 5.5 fln, the ratio o f f ( r ) tof(oo) is about 0.1 and, at r ~ 8 fm, this ratio is about 0.65. Thus, if the channel radius a c is taken equal to the potential well radius (~'-, 5 fin), the contribution to the dipole overlap integral arises almost exclusively from the external region. For ac ~ 5.5 fm, the ratio of I(ac < r < ~ ) to I(0 < r < ~ ) is 0.86 and, for ac ~ 8 fm, this ratio becomes 0.13. Hence, the contribution from the internal region is not negligible, unless the channel radius ac is chosen close to the radius of the potential well used. We now turn to radiative capture at thermal energy. While the simple model constructed in subsect. 4.1 leads to an order-of-magnitude agreement between predicted and measured radiative widths in the keV region, its application to neutron thermal capture must, however, be considered with caution. Neutron thermal capture reactions can be strongly influenced by a negative-energy resonance, i.e. by a bound level. The parameters (position and width) of this resonance are determined, in practice, by fitting the total cross section as a function of energy, and the thermal capture cross section. The results obtained from this procedure depend upon the TABI.~ 3 Comparison of calculated and measured 7-ray intensities in neutron thermal capture by 5aNi Echo (MeV)

j~r

02

Ilj (calc.)

lfj (meas.)

0 0.466 0.878 1.303

338½-

0.688 0.625 0.080 0.270

550 300 48 98

490 210 48 17

Itj (ref.

20))

534 235 46 131

size of the available data. For instance, Moore et aL 19) fit their data for 56Fe with a negative-energy resonance at -4.39 keV, while the data of Garg et aL 14) require a negative-energy resonance at - 2 . 0 keV. We have shown in subsect. 4.1 that at very low incident energies, such as thermal energy, we have it2) ap ~ P½P fo ~ (°)~PpIHi~IIZE'>~c(E')de'

(4.19)

One can use approximations (4.4) and ~4.5) and wriie ¢2) 2p

~

Np½(O)~pplnintlZ~>'

(4.20)

where N is a factor containing the unknown neutron partial width amplitude ~c. The knowledge of N is not necessary for the calculation of thermal capture intensities. Let us consider the neutron thermal capture by 5aNi which has already been analyzed by Mughabghab 2 o) in the framework of Lynn's (R-matrix) valency neutron model [ref. a)]. The results are shown in table 3 where are listed the excitation energies of low-lying single-particle states J~ with significant spectroscopic factors 20) 02, the

RADIATIVE CAPTURE

615

calculated and the measured 2 o) thermal capture intensities. The calculated intensities, in units of photons per 1000 captures, are defined as the ratio

I,j- r~(i-.j) xlOOO,

(4.21)

E r p(i -. j) J

where i ~ j denotes the transition from the initial (s-wave) state to one of the four final statesj listed in table 3. The factor N in eq. (4.20) thus cancels out in the calculation of I~j. Examination of table 3 indicates that the calculated and measured intensities are in good agreement except for the state at 1.3 MeV. The last column of table 3 shows the results of the R-matrix approach 2o); they are close to the results obtained f r o m the shell-model approach. We note that when only one particle channel contributes to the part ,~2) of the photon width amplitude, an asymmetric shape is obtained for the resonance, the cross section vanishing at some point. This was emphasized in ref. 1o). Using approximation (4.4), one finds that ,~s.m. 2 (E-Ez+Ax) 2 a ~ oc -cp . (I) where, in the case rxp

~

(4.22)

(E_E~)2 +¼F 2 ,

O, d~ = F~c Re r(ez, ~o)/2n,

(4.23)

is the level shift of the resonance due to its coupling to the continuum. This phenomenon arises from the fact that ,t2) is complex. This remark may be of importance when the value of the background cross section is obtained f r o m the asymmetry of a resonance peak 21--23), and also when the partial width F~.p is obtained from the measurement of the area under a resonance, which is usually assumed to have a Breit-Wigner shape. Finally, we mention that the energy dependence of Re F is too small to introduce any sizable correction to Fxp, from the energy dependence of E z. We are grateful to A. M. Lane for stimulating discussions, and to J. Cugnon for computational assistance at the initial stage of this work.

References 1) 2) 3) 4) 5) 6) 7) 8) 9)

A. M. Lane and J. E. Lynn, Nucl. Phys. 17 (1960) 563 A. M. Lane and J. E. Lynn, Nucl. Phys. 17 (1960) 586 J. E. Lynn, Theory of neutron resonance reactions (Clarendon Press, Oxford, 1968) M. A. Lone, R. E. Chrien, O. A. Wasson, M. Beer, M. R. Bhat and H. R. Muether, Phys. Rev. 174 (1968) 1512 A. M. Lane, Ann. of Phys. 63 (1971) 171 S. F. Mughabghab, Conf. on study of nuclear structure with neutrons (Budapest, 1972) in press C. Mahaux and H. A. Weidenmiiller, Shell-model approach to nuclear reactions (North-Holland, Amsterdam, 1969) E. Boridy and C. Mahaux, Nucl. Phys. A209 (1973) 604 E. Boridy and C. Mahaux, Phys. Lett. 45B (1973) 81

616 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23)

E. BORIDY A N D C. M A H A U X C. Mahaux and A. M. Saruis, Nucl. Phys. A138 (1969) 481 Nguyen Van Giai and C. Marry, Nucl. Phys. A150 (1970) 593 A. A. Ross, H. Mark and R. D. Lawson, Phys. Rev. 102 (1956) 1613 H. E. Jackson and E. N. Strait, Phys. Rev. C4 (1971) 1314 J. B. Garg, J. Rainwater and W. W. Havens, Phys. Rev. C3 (1971) 2447 B. L. Cohen, R. H. Fulmer and A. L. McCarthy, Phys. Rev. 126 (1962) 698 E. G. Bilpuch, K. K. Seth, C. D. Bowman, R. H. Tabony, R. C. Smith and H. W. Newson, Ann. of Phys. 14 (1961) 387 M. R. Bhat, R. E. Chrien, S. F. Mughabghab and O. A. Wasson, Statistical properties of nuclei, ed. J. B. Garg (Plenum Press, New York, 1972) cont. 4.1. A. M. Lane, Neutron capture gamma-ray spectroscopy, Proc. Studsvik Symp., Aug. 1969 (IAEA, Vienna, 1969) J. A. Moore, M. Palevsky and R. E. Chrien, Phys. Rev. 132 (1968) 801 S. F. Mughabghab, Phys. Lett. 3513 (1971) 469 C. M. Shakin and M. S. Weiss, Phys. Rev. C2 (1970) 1809 R. J. Baglan, C. D. Bowman and B. L. Berman, Phys. Rev. C3 (1971) 672 H. E. Jackson and R. E. Toohey, Phys. Rev. Lett. 29 (1972) 379